Sunday, May 9, 2010

This review contains spoilers near the end. I will include a second spoiler warning right before they start.

There seems to be an unwritten law among mathematicians that we must all be head over heels in love with Neal Stephenson and in particular his 2008 novel Anathem. Having read the novel I find I am left with no option but to become an unwritten outlaw.Stephenson’s popularity among mathematicians is understandable. When faced with movies like The Number 23, the thought of a successful work of fiction by someone who has some actual understanding of math comes as a breath of fresh air. Also, Stephenson is a talented writer. His prose finds a perfect midpoint between beige and purple and he does an amazing job at pacing his novels so that even at 900+ pages they never drag.

But Anathem isn’t just a novel. It’s a great novel of ideas that immerses the reader in a richly detailed alternate world. But this begs certain questions. What good is a great novel of ideas when the ideas are garbage? And what good is immersing the reader in a richly detailed alternate world when said world sucks donkey balls?

Those familiar with Neal Stephenson know he is a right-wing libertarian, with the usual right-wing libertarian fetishes (like the gold standard and pretending that small undemocratic tax-haven countries represent the pinnacle of freedom). This acts as a strike against all of his novels, but usually he’s able to overcome these weaknesses by talking about math.

Anathem appears at first to be about math, which should play to Stephenson’s strengths. But, it turns out it’s really about the philosophy of math, which means it’s completely dominated by Stephenson’s elitist views. On top of that we get a ginormous dose of new-age mysticism.

Platonism

Stephenson has said he wrote Anathem to explore the ideas of Platonism. He claims to stand in the tradition Plato, Leibniz, Kant, Russel, Gödel and Turing.

I have dealt with Platonism previously in greater detail, but I will recap. Platonism is the belief that the world in which we live is a shadow of another world, the Platonic Ideal (referred to as the Hylæan Theoric World in the novel). For instance in our world, there are no perfect circles, only objects that look like circles but, on closer inspection, turn out to have jagged or warped edges. In the mythical Platonic Ideal there are genuine perfect circles and our garbage can lids are just shadows of those mystical objects. It’s an interesting philosophy that suffers somewhat from the fact that it’s not true.

Closely related to Platonism is formalism, which also says that the real world in which we live is a shadow of another world, but which states that this other world is inside our head. According to formalism, a garbage-can lid in our world is just a game-piece for us playing a game with the axioms of Euclidean geometry in our head. Like Platonism it’s an interesting philosophy that suffers somewhat from the fact that it’s not true.

Historically, these two philosophies have been presented as polar opposites despite being very similar to each other. As such, many mathematicians who get turned off by the mysticism of one of the philosophies have aligned with the mysticism of the other philosophy. As such most people you meet in real life who identify as Platonists or formalists are perfectly reasonable people. Not Neal Stephenson, however.

In Anathem the mathematicians are completely divided into two warring camps: the Platonists (Halikaarnians in the book) and the formalists (Procians in the book). The Procians are all portrayed as either evil or stupid, and often both. Also, every mathematician identifies with one of the two philosophies, so there is no one who takes any form of materialist view of math in the entire world of the novel.

Despite Stephenson’s props to Russel, Gödel and Turing, he takes an incredibly extreme view of Platonism that those three would shudder from.

Russel, Gödel and Turing were rationalists who mostly took a materialist view of the universe. When it came to the specific question of where the rules of reason came from, they slipped into Platonist mysticism. This was a defining characteristic of enlightenment philosophy (like Kant): arguing against religion in favor of reason and materialism, but taking a mystical view to where reason comes from.

Stephenson, on the other hand, is fare more in line with Henri Poincaré’s interpretation of Platonism that argues against reason in favor of intuition. He also holds certain truths to be self-evident that definitely not self-evident, including some that are demonstrably false.

Poincaré argued that space-time had to be Euclidean because Euclidean geometry was the simplest model of mathematics, and hence the truth. Stephenson takes this to an even crazier extreme, where he insists that certain coordinate systems are universally true while other one’s that model the exact same thing aren’t.

Beyond the mysticism, Stephenson’s Platonism has an arrogant elitism mixed in, that goes hand in hand with his right-wing libertarianism. The formalists are evil and/or stupid because they just don’t have what it takes to be a real mathematician like the Platonists. The same goes for those outside of the math world.

The result is that he adopts all of the obnoxious of the most stereotypical atheists and uses it to argue for mysticism.

Setting and Made-Up Words

For those who haven’t read Anathem, it is a science fiction novel set in an alternate universe, on a planet where all the mathematicians are kept isolated from the rest of the world, where they can devote all their time to their work.

Because it’s set on an alien world in another universe, the characters don’t speak English, but rather made-up languages of Orth (for the mathematicians) and Fluccish (for everyone else). To work around this, Stephenson writes mostly in English with some made-up words tossed in there, to give the setting an alien feel while still being comprehensible.

He says in an introductory note “Rather than use the Orth word, which would be devoid of meaning and connotations to Earth readers, I have tried to devise an Earth word that serves as its rough equivalent while preserving some flavor of the Orth term”.

This mostly reads well and it’s not hard to figure out what most of the made-up words mean (gee, I wonder what “theorics” and “praxis” mean). However, throughout the book there are entries from The Dictionary, some of which are just awful.

When characters say “bulshytt” it’s fine. When the word “bulshytt” gets a long dictionary definition that treats it like a technical term, it’s awkward. When that dictionary definition includes a side comment about how non-math people view the term as an insult but math people view it as a technical term but they can’t be sensitive to the non-math people’s wishes without engaging in the bulshytt, it’s obnoxious. When the side comment in that dictionary definition includes speculation that this difficulty concerning the use of the word “bulshytt” is responsible for the conflicts between math people and non-math people, it’s just stupid. I really hope people don’t start using Stephenson’s spelling of “bullshit” in real life.

But even without the definitions being given some of the made-up words are, in and of themselves, troubling. Namely, those that portray mathematics as a religion.

Math as Religion

So the mathematicians’ isolated world is called the Mathic world, and it’s modeled off of a medieval Catholic monastery run by people who can’t spell. The Mathic world is made up of scattered “concents”. The people there wear monk-robes and chant and have a lot of rituals. The men are called “fraa” and the women are called “suur”. Regardless of their gender they are called “avout”. Really important people are called “saunts” and are depicted in stained-glass windows. The rest of the world is called the “Sæcular” world.

Given Stephenson’s claim that his made-up words are based on Earth words that serve as a “rough equivalent”, one would think it would make more sense to use a word like “univærsitie” instead of “concent” or “Lib’rahl Ærts Majœrr” instead of “sæcular”. The only explanation is that Stephenson views math as a form of religion.

Adding to this, we learn of the story of Deät and Hylæa, two ancient historical figures. They were said to have been the daughters of some buy named Cnoüs, who claimed to have had a vision of another world.

Deät interpreted this other world to be heaven, and claimed his vision signified that we must not engage in idolatry. Hylæa interpreted this other world to be the Hylæan Theoric World, and claimed his vision signified that it’s important that we must not mistake depictions of triangles for the actual Platonic pure triangles that live in the Hylæan Theoric world.

Supposedly this was the big separation between the Mathic world, who followed Hylæa (at least until the eeevil Procians came along) and the religions of the Sæcular world, who followed Deät.

Incidentally, why are you calling it the Sæcular world and than making it so dominated by religion? Whatever happened to the whole “roughly equivalent” thing? I guess things can be considered roughly equivalent to their opposites if you use rough in the broadest sense possible. Or if you’re full of shit.

The narrator (and author surrogate) Fraa Erasmas, tells this story to a crowd of Sæculars and argues that, really, Deät and Hylæa’s two interpretations of the visions are essentially the same, so there shouldn’t be so much conflict between the Mathic world and the Sæcular religions.

All this would seem to imply that both Stephenson and Erasmas view math as being a form of religion, but when the eeevil Procians accuse the Halikaarnians of holding that view, Erasmas and the other author surrogates get all indignant. They don’t actually argue that it’s not true, they just get pissy and point to various technicalities that don’t actually prove anything. The best response they can come up with is “How dare you!”

Libertarians seem to really like making this sort of argument. How dare you accuse Milton Friedman of supporting Pinochet! He was Pinochet’s economic adviser, not his political adviser! How dare you call Ron Paul a racist. The article that said “if you have ever been robbed by a black teen-aged male, you know how unbelievably fleet-footed they can be” that was attributed to Ron Paul and appeared in the Ron Paul-owned newsletter called the Ron Paul Freedom Report was ghostwritten, so it obviously doesn’t reflect Ron Paul’s views!

It’s the same nonsense here. How dare you accuse me of equating mathematics religion. All I did was equate mathematics with religion but I also like invoking Occam’s Razor ("Gardan’s Steelyard" in the novel) against people I disagree with despite regularly violating it in my own arguments!

The Eeevil Procians

In fact, when dealing with the Procians, he barely engages in any actual discussion. Instead of that, he goes out of his way to make the Procian characters as unlikable as possible. They’re eeevil and stupid. They’re always scheming to try and tear the Halikaarnians down. They try to suppress the Halikaarnians and keep them from getting recognized.

In real life the biggest conflict between Platonists and Formalists was between Poincaré and David Hilbert. This was because Poincaré, the Platonist was trying to suppress people he disagreed with from being recognized. All the Platonist mathematicians in whose tradition Stephenson claims to stand, Russell, Gödel, Turing, sided with the eeevil Procian Hilbert against Poincaré.

The main Procian in Anathem is Fraa Lodoghir who also makes racist comments and criticizes the cooking of the Halikaarnians in order to put them on the defensive. Yes, according to Neal Stephenson, the policy of “negging” comes, not from the “Mystery™ Method of Seduction”, but from mathematical formalism.

Stephenson makes no effort to explain either how the bad behavior of the eeevil Procians stems from their philosophical views or how their philosophical views stem from their bad behavior. They’re just eeevil for the eeevil’s sake.

In fact, the Procians aren’t just straw men for formalist arguments: they’re universal straw men. Whenever Neal Stephenson disagrees with something, he makes the Procians agree with it, even if it has nothing to do with mathematical formalism. So the Procians accuse the Halikaarnians of passing on private property from generation to generation, because Stephenson supports that practice. Fraa Lodoghir has the title of “First Among Equals”, because not believing in the Hylæan Theoric World apparently causes you to engage in Orwellian double-speak.

The Procians as universal straw men thing gets incredibly absurd when it comes to issues on which Stephenson’s opinions are vague. When a businessman makes racist remarks it’s considered to be saying it as it is, but when Fraa Lodoghir makes racist remarks it means he’s an eeevil Procian.

When the Halikaarnians associate with Sæculars it means they’re taking a stand against the corrupt Procian-dominated discipline. When the Procians associate with Sæculars it mean’s they’re greedy. When Halikaarnians insist on isolation from the Sæcular world, it means they’re committed to the Hylæan Theoric World. When Procians insist on isolation from the Sæcular world, it means they’re trying to keep the Halikaarnians down.

Most absurd are the bits dealing with logical fallacies. In real life, people make logical fallacies all the time. It’s a sign of muddled thinking. Sometimes it’s because you don’t know better and sometimes it’s because you slipped up. Sometimes, logical fallacies still have some scientific validity. Post hoc ergo propter hoc is a logical fallacy, but it still counts as legitimate evidence in the scientific method. Just not conclusive evidence.

But according to the glossary, we encounter the term “Hypotrochian Transquaestiation,” which is defined as:

Only one of a very large number of rhetorical tactics drilled into fids, particularly those under the tutelage of Procians. It means to change the subject in such a way as to assert, implicitly, that a controversial point has already been settled one way or the other.

So, you see, the Procians are taught to use logical fallacies. Meanwhile, whenever the Halikaarnians engage in Hypotrochian Transquaestiation, it’s never mentioned unless they’re arguing against other Halikaarnians whose views more closely resemble those of Neal Stephenson.

Real Life Procians

While mathematical formalism is a flawed belief, it’s real-life practitioners are generally not eeevil or stupid or manipulative. Everybody recognizes Einstein as a genius. Most mathematicians regard David Hilbert in the same light, even if he is less well known outside of the mathematical community. If you take a look at his Wikipedia page you will find a large number of things named after him.

G. H. Hardy was not only not a racist, but he was the first person to recognize the genius of Srinivasa Ramanujan who other mathematicians, due to their racism, refused to take seriously. He is not known to have ever criticized people's cooking in an attempt to put them on the defensive.

Also, in the debates between Hilbert and Poincaré, Hilbert was the one who was emphasizing the importance of rigorous proofs. If you, as formalists do, believe that mathematics is just a game, what’s the point in doing it if you don’t follow the rules. Poincaré, on the other hand, was more fond of accepting things as true based on intuition, hence intuition. As such, the formalists were much more careful than the Platonists in avoiding logical fallacies and under no circumstances instructed their pupils to employ them.

The fact that Neal Stephenson prefers to simply employ mathematical formalists as universal straw men instead of actually taking on their views also means that he misses out on some legitimate criticism of formalism. Oddly enough, some of the biggest flaws with formalism are things that Stephenson himself engages in.

For instance, Formalism, when taken to an extreme, would imply a solipsist worldview in which individuals make their own truth. Since Platonists believe in an absolute universal abstract truth, they can avoid this particular bit of mysticism. And yet when we get to Fraa Orolo’s Theory of Consciousness™, he takes the formalist position.

Similarly, we have the philosophy of “sconics”. This was based on the philosophy of positivism, which included both Platonists and formalists, but specifically the variety practiced by the Vienna circle. The Vienna circle included the Platonist Gödel as a member, but was completely dominated by formalists and claimed to stand in the tradition of Ludwig Wittgenstein, one of the nuttier formalists.

The Vienna circle and the sconics were mostly materialists, but they took such an extremely narrow view of what constituted material (stuff that could be perceived by the human mind) that they ended up being agnostic about pretty much everything that they weren’t solipsist about.

In practice, both the Platonists and the formalists are usually able to ignore the kookier aspects of their beliefs when they did their work. They only engage in philosophical holy wars when they’re fighting over a grant.

Dialog(ue)

The over-the-top portrayal of the differences between Procians and Halikaarnians seriously interferes with the readability of the book, because a large portion of the dialogue in the book is Crichton-speak, referred to in the book as “dialog”. For those who don’t know, Crichton-speak is a style of fiction dialogue associated with Michael Crichton in with an author surrogate explains the author’s position while another character either asks questions or raises objections that the author surrogate will ultimately explain.

In the case of Michael Crichton, and this book, the author surrogates are unimpeachable and the other characters are either naïfs who quickly come to accept the author’s position or incredibly stupid and/or eeevil people who make terrible objections because the author finds them easier to reply to than the legitimate objections.

Of course Michael Crichton didn’t invent Crichton-speak and if you’re going to engage in polemics in a work of fiction this sort of thing is inevitable. Stephenson was likely inspired by the ancient Greek Socratic Dialogues, specifically those of Plato. Also, to a lesser extent, Galileo’s Dialogue Concerning the Two Chief World Systems.

The problem with Crichton-speak, is that it’s only as strong as the author’s opinions. In the case of Galileo, the Dialogue spans four days, three of which are great because they deal with things like the Earth not being the center of the universe. The fourth day, however, was arguing against the “crazy” notion that the tides were caused by the moon, something which turned out to be completely right.

Anathem not just because of the over-reliance on Crichton-speak, but because of Stephenson’s belief that Cirhcton-speak is the highest form of human communication (despite the fact that it only occurs in works of fiction). In the concents this is the primary means of learning. Avout are expected, not only to study famous dialogs, but to memorize them. When a “Dialog” occurs during the novel, crowds gather round to watch and maybe root for one side, but don’t participate.

Real mathematicians don’t study ancient Michael Crichton novels or replicate them. We study from textbooks and math papers and notes from lectures and seminars. Oh yeah, and we have these things called lectures and seminars, not to mention discussion groups and conferences and bar nights every other Thursday. If we did replicate Michael Crichton novels we wouldn’t gather around to watch and see who wins, because we’d know that the winner will always be the one who agrees with Michael Crichton.

Oh yeah, and when you lose an argument in a Dialog, the winner is said to have “planed” you. One of the characters, Barb, a precocious brat who Fraa Erasmus takes under his wing, develops “planing” as a hobby, especially once the characters enter the Sæcular world. As you can guess, the character manages to make Wesley Crusher seem non-annoying in comparison.

In The Dictionary, they mention three types of “Dialog”: Peregrin, in which two people “of roughly equal knowledge and intelligence develop an idea by talking to each other, typically while out walking around”, Periklynian, in which each person tries to “plane” the other, and Suvinian, in which a teacher tries to impart knowledge to a student.

Of these, only the Peregrin is in any way egalitarian and, as you can guess, it’s the type of Dialog that occurs least frequently in the book despite being the kind that occurs most frequently in real life. Even when Fraa Erasmas and his mentor Fraa Orolo meet “as equal” in the middle of the book, all their Dialog is of the Suvinian variety.

The result is that there is a hierarchical atmosphere throughout the book where every character is either an adversary a guru or a student and no one works together.

On top of that, there is this notion of a “convox”, where all the different concents get together to have a conference. This happens rarely, once every thousand years (for the sole purpose of updating The Dictionary) and in other special occasions like the events of the book.

Not only are they rare (despite the fact that math conferences are incredibly frequent in real life), but they are looked down upon by the characters in the book because large numbers of people supposedly can’t accomplish anything and it all devolves into “politics”.

This makes me wonder if they were ever able to solve the classification of finite simple groups in this universe. In this world, it was only through intense collaboration among a large number of mathematicians, to the point where there is no one mathematician who can be attributed to being “the one who solved it”. Maybe there’s a stained glass window somewhere of a bunch of people groveling at each other’s feet.

Life in the Conv(c)ent

In fact most of the concent life makes me wonder how they are able to accomplish anything. The first third of the book is pretty much devoted to describing what it’s like in the concent in the hopes that the readers can immerse themselves in the world. It’s an admirable goal incompetently executed.

The concent itself is walled off from the Sæcular world. The avout are divided into groups based on how often their allowed to leave the walls. The Unarians can go out once a year, the Decenarians once a decade, the Centenarians once a century and the Millenarians once a millennium.

There is some implication that the longer you go without visiting the Sæcular world, the smarter or better you’re supposed to be, but the main Millenarian character, Fraa Jad doesn’t come off as particularly smart. He doesn’t even believe in time (but he does believe in the Hylæan Theoric world).

Also, different concents aren’t allowed to contact each other except during the extremely rare convoxes, once again bringing in to question whether these people would be capable of solving the classification of finite simple groups.

The concents are independent economic and governmental entities from the rest of Sæcular world. This makes the complete lack of democracy somewhat disturbing. It’s bad enough that someone like Larry Summers can be made president of Harvard without having to be elected, but think of if Harvard was an independent country.

The concents are ruled by hierarchs who are appointed by the old hierarchs, and there’s an inquisition. You would think, given how much the concents resemble medieval Catholic monestaries that the inquisition would be bad guys. But their actually the “good guys” because they’re Halikaarnians whereas the hierarchs are Procians.

The hierarchical stuff is taken to an extreme by the fact that the best mathematicians are given stained glass windows that show other lesser mathematicians groveling at their feet. One of the characters, Fraa Jesry, claims that these lesser people only proved lemmas that the Saunts used to prove theorems. Typically mathematicians prove multiple theorems with each paper. And they prove their own lemmas, which is usually where most of the work is.

Most of the time seems to be spent engaging in Dialog or chanting, but there is occasional reference to lectures. We even encounter an actual lecture. Is it about math? No, it’s about the “iconographies” that the stupid extras apply to the avout because they’re all too stupid to truly understand the avout.

One thing that the avout do not do is laugh. They are the most humorless people imaginable. This is not to say that they don’t tell jokes, which they do a lot, but every time it happens, Stephenson goes out of his way to point out that the person who told the joke was either (a) oblivious to the fact that nobody found the joke funny, or (b) embarrassed by the realization that nobody found the joke funny.

In real life, mathematicians tell jokes and laugh at them, even the puns. This is because math jokes are the highest form of humor, followed by sex jokes, precisely the two types of jokes the characters fail to find funny. The lowest form of humor is “Did you ever notice that such-and-such a group walks like this but such-and-such other group walks like this?” In Anathem, this is done with avout and extras, but its not a joke, but a profound truth.

The first time extensive laughing occurs in the book is two thirds of the way through and, of course, it’s because the eeevil Procians are laughing at the Halikaarnians during a Dialog as a sleazy means of putting them on the defensive.

Eventually the avout have to “pick a major”, but rather than picking which type of math they want to do, they have to pick whether they will be Halikaarnians or Procians. And yes, there is a scene where the eeevil Procians get Fraa Erasmas drunk on wine in an attempt to trick him into joining them.

When they do this they are joining one of three groups, the Edharian Order (Halikaarnian), the New Circle (Procian), and the Reformed Old Faanians (slightly less eeevil Procians). These groups only accept a certain number of new members each year, so people will often have to join a group that’s different than what they wanted.

In real life, people may want to do number theory, then find the number theory professors aren’t accepting new students, so they do algebraic geometry instead. But number theorists and algebraic geometers aren’t in a philosophical holy war so it’s not that big a deal. When you really want to work for the side of good, but they’re not accepting any new members so you have no choice but to side with eeevil, things get a bit sketchy.

We meet a lot of “charmingly quirky” characters. There’s Fraa Jesry who really wants to be famous. There’s Fraa Lio who wants to be a ninja but isn’t very good at it and keeps getting beat up whenever he tests his skills out on his friends (I did find this genuinely charming until the real ninjas showed up). There’s Suur Tulia, who has boobs. There’s Suur Ala, who also has boobs and wants to establish a dictatorship. There’s Barb, who has got to be the single most obnoxious character in all of fiction. And there’s Fraa Arsibalt who’s just sort of there and doesn’t have any distinguishing characteristics. Oh, and there’s Samaan who is a computer guy who’s not actually part of the concent but is occasionally called in to fix computers because the avout aren’t aloud to use computers.

But beneath the charming quirks, all of these characters, as well as Fraa Erasmas the narrator, are jerks. This was probably not intentional, but given that they always talk in Crichton-speak, that’s how they come off. The bad guys also come off like jerks, because Stephenson has to make them eeevil so people will know their bad. The result is that all of the characters are completely unlikable.

If people do something bad, they get expelled from the concent through the titular anathem ceremony. In keeping with the conspiratorial paranoia that results from having two sides of a holy war having to live together, it’s implied that pretty much all of the anathems are the result of eeevil Procians trying to keep the Halikaarnians down, and when characters are anathematized, it’s supposed to be a tragedy. The attempts to portray it as a tragedy fail completely because not only are the characters unlikable but there is no convincing reason why anyone would want to live in the concents.

The early parts of the book aren’t all bad. Stephenson does a good job at describing the architecture of the concent. As such it’s possible for the reader to get immersed in the buildings, just not in any of the activities that go on in the buildings.

But this is a great novel of ideas, not a great novel of architecture. And I can’t help but find myself far more interested in how these people could possibly solve the classification of finite simple groups than anything that actually happens during that first third.

Avout vs. Extras

Another thing I wonder about is if they were ever able to solve the Four-Color Theorem or the Kissing Problem. The former says that any map can be colored with four colors and have no two countries bordering each other be the same color. The latter says that a ball can have at most twelve other balls of the same size touching it at once.

Both of these problems had so many special cases to check by brute force that no human could possibly test them all, so they had to be done with computers. The use of computers for doing proofs was controversial when it was introduced in real life.

In Anathem, the avout are forbidden by the Sæcular authority from using computers altogether, at least after the Second Sack. This brings up some interesting questions: Did the avout have a computer-based proof of the Four-Color Theorem and the Kissing Problem before the Second Sack and have to spend the rest of their history knowing the answer but not being able to prove it? Of course Neal Stephenson devotes precisely zero of the 900+ pages to even considering this question.

In general the relationship between the avout and extras is problematic, mostly due to Stephenson’s elitism. Supposedly there were three “Sacks” in the past 3689 years when the Sæcular world invaded the concents.

The first one was allegedly because the extras felt intimidated that the avout were using nanotechnology. The second was allegedly because the extras felt intimidated that the avout were using computers. The third was allegedly because the extras felt intimidated that the avout were staring at copper bowls real hard.

The third Sack was specifically inspired by these myths about sorcerer mathematicians popular amongst the Extras. They concerned “Incanters” and “Rhetors” who were supposed to be the Halikaarnians and Procians. The Incanters/Halikaarnians can will things into existence by chanting and the Rhetors/Procians can change the past. What these myths have to do with Platonism and formalism is never explained.

Much is also made of the “iconographies” that the extras have of the avout. But by presenting the extras as being too dumb to comprehend what the avout do, they’re creating their own “iconographies” of the extras. Aside from wearing funny clothes and being horrible, horrible people, the avout aren’t that different from the extras.

Despite the fact that none of the avout actually come off as particularly intelligent, they are forbidden from procreating (via birth control in their food), because the extras are worried that they’ll be bred into a hyperintelligent race.

The only way they get more avout is by Sæcular’s abandoning babies at their door.

Also, every once in a while, the Sæcular world needs avout to do things for them, like when they thought an asteroid was about to hit the planet, so some avout are “evoked” from the concent. And then they stay evoked, and yet there is no sign that the evoked avout ever make any attempt to set up schools in the Sæcular world.

Supposedly this is because the Sæcular world is pissed of that there was a nuclear war 3689 years ago that was only possible through science, which was only possible through mathematics. I can see why this would result in anti-science hysteria immediately after the fact, but they’ve had 3689 years to cool off. Why don’t they allow the avout to mingle with them? And, more importantly, why don’t the avout want to leave the awful, awful concents?

The S(a)ecular World

Well Fraa Orolo, Erasmas’s mentor, explains his decision to stay in the concent after going extramuros during that brief period during which mingling is allowed.

For a while I was quite seduced by it. Then one evening I had an especially lively discussion with a sline who was as bright as anyone within this concent. And somehow, toward the end, it came out that he believed that the sun revolved around Arbre. I was flabbergasted, you know. I tried to disabuse him of this. He scoffed at my arguments. It made me remember just how much careful observation and theorical work is necessary to prove something as basic as that Arbre goes around the sun. How indebted we are to those who went before us. And this got me to thinking that I’d been living on the right side of the gate after all.

Boy, when he meets Fraa Jad, the guy who doesn’t believe in time, he is gonna be soooo pissed that he stayed in the concent.

Also, why doesn’t he just team up with those evoked avout? Or the anathematized avout who were wronged by the eeevil Procians?

Well, in general, all of the explanations for not wanting to live in the Sæcular world amount to “because the Sæculars are stuuupid”. Just as Stephenson has to go out of his way to make the Procians extra horrible because of how unlikable the Halikaarnian characters are, he goes out of his way to make the Sæcular world extra horrible because of how repulsive the Mathic world is.

In the Sæcular world, they speak Fluccish instead of Orth. Fluccish doesn’t have a written language because the extras are all illiterate anyway (despite the fact that they all have PDAs).

The only people who respect (aside from the racist sled-transport owner who leaves people for dead once he’s been paid) are the religious people. There are four main religions in Anathem, three of which have parallels in our world. There’s Bazian Orthodox, which is based on Catholocism, Counter-Bazian, which is based on Protestantism, and Kelx, which is based on Linday Weir’s stoned ranting from that one episode of Freaks and Geeks.

Then there’s also the followers of the Warden of Heaven, who’s exactly like the Halikaarnians, but eeevil and stuuupid. They never say exactly what makes him different, and he’s mainly used as “proof” that the Halikaarnians don’t view math as a religion, you know, because they make fun of him and stuff.

The fact, that the other three religions are treated with such reverence including an obnoxious scene where the avout bond with a bunch of Bazian monks over their common love of chanting, doesn’t bode well for your claim that you don’t view math as a religion.

Oh, and if you’re making the made-up words have real-life parallels, why didn’t you make the Sæcular World be secular?

When the avout go into the Sæcular world they’re all disappointed by how boring it is, which makes no sense because life in the concent is so boring that anything would be exciting in comparison (it also makes no sense because all of the most exciting parts of the book take place in the Sæcular world).

Stephenson goes out of his way to present the Sæcular world as being full of unnecessary complications like PDAs and unions. But PDAs and unions aren’t complications they make life better. If I was living in the Anathem the PDAs and unions alone would be enough to convince me to live in the Sæcular world.

The stuff about the PDAs (“jeejahs” in the book) is particularly silly. When extras are visiting the concent while the doors are open, Fraa Erasmas is talking to them about the Hylæan Theoric world, and one of the extras asks what the Hylæan Theoric World would be like. Then a jeejah rings, and Erasmas says “for one thing, there’ll be no jeejahs”. This an a later thing where they complain that the Graphical User Interfaces on the jeejahs makes things more complicated, and it makes all of the avout come off like crotchety old men demanding that the Sæcular world gets off of their lawn.

The biggest disappointment with the Sæcular world comes from the fact that it has made no progress in the past 3689 years. It’s said to just keep shifting back and forth between various forms of government, never making any big improvements. And technologically, it’s only slightly ahead of where we are today.

There was even a “dark age” of the internet, where the internet was completely worthless. There is a reason given for this but it’s so implausible that it makes the “shoe event horizon” from The Restaurant at the End of the Universe seem like common sense. It seems as if Stephenson overcompensated for his “the internet will make everything obsolete” view from his earlier books and has now rapidly switched to luddism.

Pangloss the Pessimist

Stephenson does give a (terrible) explanation for the lack of development, which is like a more pessimistic version of Leibniz’s Panglossianism. For Stephenson this is the best of all possible worlds, not because it’s good, but because we don’t have the brain capacity to live in a better world.

He gives the example of a gas stove, and how it’s inefficient and dangerous, but easy to fix. A fusion-powered stove (it’s science fiction, remember) would be safer and more efficient, but you would need to have advanced knowledge in physics to fix one, and that’s why everybody uses gas stoves.

To give an example of my own, the human body is really complicated, so naturally if it breaks down, you need to have advanced knowledge in the biology of humans, the chemistry that drives that biology, the physics that drives that chemistry and the mathematics that drives that physics. So if you get a cut, you better find some super-genius or you’ll never know to sterilize it with alcohol before putting a band-aid on it. Clearly we should just abandon humans for being too complicated.

While it may take a lot of thought to invent stuff, it doesn’t take as much thought to make them once the blueprints are there, and it doesn’t take much thought to fix them. This is because to invent things you need to know how things word, but to make and fix them you only need to know that they work. Nobody has a complete working knowledge of the universe, but between us we have enough knowledge to get through life.

In the case of stoves, you don’t need to know the chemical differences between propane and butane to fix them (for the record, propane is C3H8 and butane is C4H10; they have similar chemistry, but butane, being bigger, has a higher boiling point), but you do need that knowledge to design a gas oven.

Even if there is a limit to what the human mind can comprehend, you’d think it would be a bit more advanced than what we have. Beyond the “newmatter” which was mostly (but not entirely) abandoned shortly after it was discovered, there isn’t nay form of nanotechnology.

Part of this “the human brain can’t handle any more progress” comes from Stephenson’s insane individualism, that treats pretty much all forms of collaboration as eeevil or impossible (hence the whining about how convoxes never accomplishing anything).

But even with current technology, you need large-scale collaboration for anything to work. Stephenson’s inability to grasp this not only gives him a reactionary, pessimistic view of the world, but it means that the role of the tiny Mathic world makes no sense.

Political Economy

Supposedly, the Mathic world has lived independently from the Sæcular world for 3689 years, with the sole exceptions being the “Sacks”. At the time in which the novel is set, this has reached the point where the entire Sæcular world is one united Federation with ID cards that let people travel anywhere, but the avout are the only people who don’t have IDs.

At the same time, the Mathic world is a subsistence economy. Everybody’s spending all of their time staring at copper bowls and thinking about the Hylæan Theoric World, so they never get around to producing any wealth.

The avout do have jobs, called “avocations”, like gardening which provides food to eat. There are even some people who devote all of their time to their avocations rather than staring at copper bowls. But they are viewed by everybody else as objects of pity, as people who couldn’t cut it with their studies. They’re said to have “fallen back”. If people devote too much time to gardening or beekeeping or cleaning then the other about start viewing them suspiciously because those are signs that someone is about to fall back. Therefore the standard of living of the entire concent is based entirely on objects of pity.

Even if everybody did actual work, the concents are tiny, and not economically self-sufficient. If there’s a drought the avout would starve while the Sæaculars surrounding the concents would be able to get food shipped in from another part of the world.

The matter is made worse by the fact that the different concents don’t even communicate except for the extremely sporadic “convoxes”, so they can’t even get food shipped from another concent.

So, in order to the concents to sustain themselves over a period of 3689 years they would either have to have the extreme good fortune of having bountiful harvests year after year for the entire duration of that time or they would have to steal from the Sæcular world. If it was the latter, that would explain what the Sacs were all about but, according to the book, the Sacks were the result of the Sæculars being stupid.

No Sense of History

The lack of a thought-out political economy extends to a lack of a thought-out history, which seriously impedes any attempts at world-building. When you try to immerse the reader into a made-up world you have to give a sense that there’s some rhyme or reason for why the world is the way that it is. Despite having a long historical timeline at the beginning and references to various historical events throughout, there is no actual sense of history.

For instance, in the holy war between the Procians and the Halikaarnians, the book portrays the Procians as being the ones in power, and the Halikaarnians as an oppressed minority. And yet all of the avout who are said to have made big accomplishments turn out to be Halikaarnians.

And if the Procians are in power, why is The Dictionary, written by the avout at the convoxes that the Halikaarnians hate so much because it involves “politics”, which is supposed to be what the Procians are all into, so skewed towards the Halikaarnians? In the definition of “Sphenics”, a group of philosophers that developed into the Procians, it says “Their most prominent champion was Uraloabus, who in the Dialog of the same name was planed so badly by Thelenes that he committed suicide on the spot.”

This is a dictionary, not an encyclopedia, so it shouldn’t be going into biographical gossip about the people involved in the movement it’s defining. Furthermore, in the glossary at the end of the book, it says that Protas, Uraloabus’s opponent claimed, that that was the reason for his suicide, so it’s not even certain that that’s what it was all about. Even if there was justification for including that information, given how eeevil and manipulative the Procians are, they could have censored it anyway.

Also note that the dictionary doesn’t say what the Dialog was about so people have no way of verifying whether Uraloabus had a point and only lost because Protas engaged in sleazy debating techniques or because Protas was genuinely right.

And now, another dictionary entry, for “Causal Domain”, presented in its entirety:

“A collection of things mutually linked in a web of cause-and-effect relationships.”

This is a book that is written in this important event once every thousand years. It’s meant to be the foundations of the work of the avout, many of whom probably do stuff in the mathematics involved in relativity. And the avout are mathematicians, who depend on rigorous definitions to do their work. And that’s the best definition they can come up with for “Causal Domain”? I guess they just needed space for the unsubstantiated rumor about Uraloabus.

There’s also no good reason why the philosophical holy war appeared on those lines, and why no views other than Platonism and formalism ever appeared, in particular why there is nobody who thinks that mathematics refers to things in their own world?

And why did the difference between Procian and Halikaarnian become more important than the difference between algebra and analysis? What horrible event caused people to think that memorizing ancient Dialogs would be a good way to spend your time? And who thought it would be a good idea to forbid avout from different concents from communicating?

And how was it that the Procians managed to end up as the ones in power anyway? I know that Stephenson needed that to be the case so the “good” guys would come off as the underdogs, but was there ever at least a period when the Halikaarnians were in power? I mean it’s set in the concent of Saunt Edhar, and Edhar is a Halikaarnian, but the Procians run the concent. Something must have cause that to happen.

There’s even less justification for the history dealing with the Mathic and Sæcular world. Supposedly the two worlds were separated as a result of unknown “terrible events” involving a nuclear war, that happened 3689 years ago, but there’s no reason for them to stay separated over such a long period of time.

Why didn’t the evoked avout start teaching math? Why didn’t people try to reinvent the wheel and start over again? It’s not like you need to live in a concent to learn math. Why didn’t the Sæcular world ever annex the concents? If the Sæcular world really got to the point where they were a single federation with ID cards, why didn’t they come up with some means to keep track of the comparatively tiny number of avout?

Why wasn’t there ever, in the entire 3689 years, a movement to end the separation of the Mathic and Sæcular worlds, even during the asteroid scare? During the time after the Sacks forbade use of computers why wasn’t there a movement among avout to revoke that?

In general, things happen in Anathem, not because there’s any reason for them to happen, but because the author is trying to make a point. The reason the definition of “Sphenics” includes the rumor bout Uraloabus is because Stephenson wants to make the point “The Procians are stuuupid!” The reason the definition of “Causal Domain” is so vague, is because there’s a “scientific” explanation given of the Hylæan Theoric World that uses Causal Domains that makes no sense if you actually understand causality. The reason for the lack of any attempts to unite the Sæcular and Mathic worlds is because, well, I don’t know.

But this makes it incredibly difficult to immerse oneself in the world in which Anathem is set.

Where’s the Math?

Of course, not everyone is reading Anathem to immerse themselves in a made-up world. Like most of Neal Stephenson’s novels, Anathem is ostensibly about math, and its success is somewhat due to its mathematical content. And yet, there is very little legitimate math in the novel. For the most part, characters are too busy thinking about the Hylæan Theoric world to busy themselves with math.

In something like Good Will Hunting the lack of legitimate math is somewhat acceptable because the movies isn’t about math per se, but about a mathematician and his life. It still would have been a better movie if it had better math, but it’s forgivable. Anathem, on the other hand is about math and the philosophy of math, so the lack of math is much less forgivable.

There is reference to a tiling puzzle called the “teglon”, which is supposed to be a great unsolved problem. Supposedly it involves tiling a regular decagon in accordance to certain rules, but the book never states what those rules are. More importantly, The Dictionary never states what those rules are, probably because they needed space for unsubstantiated rumors about Uraloabus.

As far as actual math, there is an unnecessarily complicated proof of the Pythagorean Theorem (“Adrakhonic Theorem” in the book), and there are three footnotes (“calcas” in the book) which deal with math to some degrees. In all of these cases, the math is rushed over in order to get to some speechifying about the Hylæan Theoric World.

Most of the book is philosophizing about math that is completely divorced from any actual math. Some examples of math that is not mentioned in the book:

The Axiom of Choice and the Continuum Hypothesis: These are axioms that are independent of the other axioms of set theory. There are models in which they’re true and models in which they’re false. If you’re a Platonist and believe in absolute truth that would extend to those axioms, even though there is no way to prove or disprove them from other axioms. This provides some interesting philosophical things to think about that Anathem ignores completely.

Russell’s Paradox/Gödel’s Incompleteness Theorem/Church-Turing Thesis: Why not mention the most well-known results of all those mathematicians in whose tradition you claim to stand? Especially since these things all call into question the notion of “universal truth”. Gödel’s Incompleteness Theorem says that there is always something that is true but can’t be proven. It also says that there is no way to prove a theories own consistency.

Oracle Computation: Alan Turing introduced the abstract notion of an “oracle” in computation, where you have some sequence “written down” that you can “look up” in addition to doing all the usual recursive computations. With just the recursive computations you can’t solve the halting problem, but hypothetically one could have a solution written down and make new more powerful computers, but those would have a new halting problem, which can’t be solved with the oracle. So you can make an oracle for that, which gives a new halting problem which the new oracle can’t solve. If the Hylæan Theoric World existed and behaved the way Stephenson describes it, it would presumably act as some sort of oracle, and it would be worth pondering where this oracle fits in the arithmetic hierarchy.

Complexity Theory: This is especially important given all the times Stephenson mentions the word “quantum”. It’s the theory of how fast algorithms can be computed. Given Stephenson’s new-age insistence that the human mind is more than “just” a machine and can do things that no computer will ever be able to accomplish, this would raise some complexity theory questions that are never asked, let alone answered.

Posets: See the section on DAGs.

Infinity: Is there some sort of absolute infinity? (no, there isn’t, just potential infinity, but maybe Stephenson thinks there is) If infinity exists how do you explain the fact that we never encounter infinity of anything? If it doesn’t, how do you explain the fact that infinite sets can be used to model real-life things? What does Cantor’s diagonalization argument and all the different sizes of infinity say about all this?

Ring Theory: One of the supposed “smoking guns” Stephenson uses to “prove” the existence of the Hylæan Theoric World is the claim that “three is a prime in any universe”. But three isn’t even necessarily a prime in this universe, for instance in the ring Z[sqrt(-2)], where we have 3 = (1 + sqrt(-2)) * (1 - sqrt(-2)).

Abstract Stuff in General: In modern mathematics there are lots of “grins without cats”, abstract concepts for which there is no known real-life thing represented by them, but still make mathematical sense. Examples include non-measurable sets, high-dimensional manifolds, large cardinals and germs of stalks of sheaves of schemes. All of the math discussed in Anathem is stuff that has known applications.

Model Theory: A branch of higher logic in which one looks at sets of axioms and studies how those axioms can be modeled. What happens when you can model a set of axioms that the absolute truth of the Hylæan Theoric World deems false? And what happens when you have multiple models of the same theory (which, according to the Löwenheim-Skolem Theorems happens quite a bit)?

Category Theory: To the book’s credit, it does use the word “isomorphic” once (and only once). But it never looks at Category Theory, which studies abstract “objects” with abstract “morphisms” that transform one to another. Objects are “isomorphic” if you can turn one into the other and then back into itself. This is a form of equivalence. You can have two categories with the same objects but different morphisms so that two objects will be isomorphic in one category but not in another. This sort of thing can explain how, in some sense, a garbage can lid is equivalent to a circle, but in other senses, it is not. And that can be done without appealing to any absolute truth handed down from on high through an alternate universe.

But we get none of that. Instead we get a few simple math problems with explanations that quickly stray to random mumblings about the Hylæan Theoric World. Let’s take a look.

Let Them Cut Cake

The first of these calcas is a simple high-school-level geometry problem about trying to cut a piece of cake a certain size. It has little trick using the fact that the ratio between the diagonal of a square and one of its sides is the square root of two. The fact that it’s such low-level math is understandable since Fraa Erasmas is explaining it to a kid, Barb.

But then Fraa Erasmas makes the “shocking” (not to mention “deep” and “philosophical”) remark that this technique can also be to mark plots of land that are a certain size. “It really makes you think of the Hylæan Theoric World,” he says.

OMG you can make cake in the shape of a square and plots of land in the shape of a square! Clearly there must be another universe containing particles of pure squareness! That’s the only explanation.

OMGWTF! And did you notice that lemons are yellow and bananas are yellow, but lemons and bananas are not the same thing? Holy shit! How is that possible? Clearly there must be another universe of pure yellowness that emits particles of yellowness into our universe! I call it the “Yellow Theoric World” (no, make that the “Yællœw Theoric World”). It’s the only possible explanation. How else could the human mind notice that both things are yellow if we weren’t getting signals from another universe! I mean, it’s not like we’re receiving particles in this universe called, I don’t know, “photons”. That would be crazy!

This line of silliness is the main justification for the existence of the Platonic ideal, nut just by Stephenson, but by everybody who believes in the Platonic Ideal. It was in inspiration for Aquinas’s fourth “proof” of the existence of God, which is famous for being the most flamingly stupid “proof”.

Aquinas’s version states that there are things in the universe that are described as good, but nothing encapsulates pure goodness. But without pure goodness there will be nothing to compare other good things to (why?), and therefore there must be some being of pure goodness (why?) and that being must be God (why?). Substitute “square” for “good” and “Hylæan Theoric World” for “God” and you get Anathem in a nutshell.

The One True Coordinate System

The second calca is about configuration spaces. This is a more advanced mathematical concept, but the calca never uses it for anything. It does a decent enough job at explaining what configuration spaces are and relating them to Kepler’s laws by providing an alternate coordinate system for configuration spaces that gives information about orbits.

But instead of talking about the importance of being able to go back and forth between coordinate systems, Stephenson decides that the orbit-friendly coordinate system is “better” than the traditional coordinate system because “it’s closer to the truth”. The exact conversation (WARNING: reading the conversation will want to make you strangle the characters):

“There’s got to be more to it than that,” Barb said. “We could have just plotted this in a simpler way.”

“This is simpler,” I insisted. “It is closer to the truth.”

“Are you talking about the Hylaean Theoric World now?” Barb asked, half whispering and half gloating, as if this were just about the naughtiest thing that a fraa could do.

“I’m an Edharian,” I answered. “No matter what some people around here might think…that’s what I am. And naturally we seek to express what we are thinking in the simplest, most elegant way possible. In many—no, most—cases that are interesting to theors, Saunt Hemn’s configuration space does that better than Saunt Lesper’s space of x, y, and z coordinates, which you’ve been forced to work in until now.”

Must. Resist. Urge. To Strangle. Characters.

Close one. Anyway, this is one of the biggest flaws with Platonism. The main reason why Einstein identified as a formalist was because Poincaré insisted that Euclidean geometry, as the most mathematically simple must be true by Occam’s razor, and so it should be used in all models of the universe, even though with relativity elliptical geometry makes more sense.

On top of that, let’s take a look at the specific coordinate systems Stephenson/Erasmas are talking about. What Erasmas refers to as “Saunt Lesper’s space” is a six dimensional space where you have three dimensions to tell the location of an object, and three dimensions to tell its velocity.

“Saunt Hemn’s space” also has six dimension, that talk about specific properties of orbits: “eccentricity, the inclination, the argument of perihelion, and three others with complicated names that I’m not going to rattle off now”.

The former is useful because it tells you exactly where the object is located. The latter is useful, because it makes it’s trajectory easier to plot. If you know where an object is now and what it’s velocity is, and you want to find its new location after a period of time, you need to convert from Lesper’s coordinates to Hemn’s coordinates, calculate the trajectory and then convert back to Lesper’s coordinates. You don’t discard Lesper’s coordinates just because Hemn’s coordinates are easier for a specific task.

In fact there are lots of different coordinate systems that are useful for different acts. If we’re walking on the surface of a planet, we use two-dimensional Euclidean space, because height doesn’t matter and the curvature of the planet isn’t significant enough to effect things.

If you’re traveling long-distance, you use two-dimensional elliptical geometry to account for the curvature of the planet. If you’re going by plane, you add a third dimension. If you’re going into space, then you add the three velocity dimensions, because the way the gravitational force acts on an object varies more and effects your controls.

If you’re staying in orbit around a single planet, then you use Hemn’s coordinates, because the gravitational forces of other bodies is insufficient to interfere with Kepler’s laws. But if you are going from a planet to a moon or to another planet you need Lesper’s coordinates, because Kepler’s laws don’t describe the motion of three or more bodies.

If you’re traveling long-distance at great speed you need to add in the Lorenz transformations of special relativity, and if you are going quickly by large gravitational bodies then you need a new model that accounts for the bending of the light cones that arises in general relativity.

If you go near the big bang or the big crunch then the laws of space and time get distorted that scientists today don’t know how they’re modeled. One day they may find a model, and that will go beyond the general relativity model. Plus if string theory turns out to be legit you have the extra six dimensions and if there’s a multiverse that’s potentially a bunch of other dimensions. But you still only need two to go to the store.

Of all these coordinate systems, Hemn’s coordinates occur somewhere in the middle, which makes even more odd that Erasmas would claim that they are closest to “the truth”. And strictly speaking, if there is an absolute truth, then Lesper’s coordinates would be closer. This is because Hemn’s coordinates assume that there are only two bodies.

Assuming that Hemn’s coordinates are “the truth” is like assuming that the world is flat. Strictly speaking it’s not true at all, but if you don’t move too far away then it’s close enough to true that it doesn’t matter.

Later on in the book there is a scene involving people flying around in orbit and trying to change orbits to meet each other, one of the cases in which Hemn’s coordinates make most sense. The supposed intent of this is to show that Erasmas was right and that Hemn’s coordinates are “the truth”.

But the fact is, most people don’t fly around in orbits trying to meet each other. If they had been trying to get to a moon base instead of a satellite, then Lesper’s coordinates would have been better and the Procians would have won. And if they were trying to get some place a few miles away then the slines would have won.

The real shame about this section is that there is some legitimate philosophizing that can be done about the changing of coordinate systems and models that aren’t exact but close enough for a given circumstance. But it’s completely wasted by all this nonsense about the Hylæan Theoric World.

DAG-Nabbit!

The mathematics in the third calca amounts to the following: “directed acyclic graphs sure are pretty”. This is also the sole attempt at giving scientific legitimacy to the notion of the Hylæan Theoric World and, in doing so, it replaces Platonism with a completely different theory that’s still wrong.

The calca deals with the views of “Simple Protism” and “Complex Protism”. These are different models of the Hylæan Theoric World (sort of). In Simple Protism, you have two universes, the one where the characters live and the Hylæan Theoric World. And there is a causal connection from the Hylæan Theoric World to the universe the characters live in. This means events in the Hylæan Theoric World effect events in the main universe, but not the other way around.

In Complex Protism, you have a whole bunch of universes, with various causal relations. The one restriction is that the causal connections have to form a directed acyclic graph (DAG). This means you can’t have a situation where Universe A effects Universe B which effects Universe C which effects Universe A again. This is in order to preserve causality. This means that there are a bunch of Hylæan Theoric Worlds and our own world may appear as the Hylæan Theoric World to some other world.

For non-math people, a directed acyclic graph is a network of dots connected with arrows (directed graph) such that there are no cycles (acyclic).

There are just so many things wrong with this theory. And there are so many things wrong with the way it’s presented.

First of all, Complex Protism is not Platonism, even though it’s presented as such. It’s an entirely new theory. Like Platonism and Formalism, it presents truth as coming from out of this universe (as opposed to materialism). But Platonism believes in a universal truth, so, unless the DAG is a path (as in the “Freight Train” model) it’s not Platonism, and all the ramblings about Hemn’s coordinates being “true” and Lesper’s coordinates being “false” ceases to hold water (not that it ever did to begin with).

So there are certain DAG’s that will produce a Platonist model, but there are others (like the “Firing Squad” model) that are much closer to formalism. Essentially formalism says that each person has their own Hylæan Theoric World in their heads, and that can be modeled by a DAG in complex protism.

In religious terms, imagine what would happen if the pope announced his new theory of God that went as follows: there is not a single God, but a whole bunch of beings, some of which are so powerful that the appear as Gods to us just as we appear as Gods to amœbas. Do you think the cardinals would go along with this?

It should be noted that this calca occurs a third of the way through the (extremely long) book. It takes another third before anybody talks about Complex Protism again, and in the (extremely long) intervening time still referring to the Hylæan Theoric World not a Hylæan Theoric world or the Hylæan Theoric Worlds. They just ignore their own theory until it’s convenient to make terrible arguments against the Procians.

Now, let’s say you’re a materialist like me, and you think we don’t get our information from other universes, but from the universe we’re in. Is there any model for that? Yes, exactly one, the one with one universe. But there is no materialist model of a multiverse consistent with Complex Protism. This relates to the confusion throughout the novel concerning what causality is.

Normally, we don’t talk about a causal connection from one universe to another, but from one event to another. An event is a single point in space-time. There is a causal connection from one event to another if an object can travel from the first point to the second. Under the Galilean model of space-time, this would mean that the first event occurred before the other.

Under Einsteinean relativity, you have the added issue that nothing can travel faster than the speed of light, so it means that the second event is in the upper half of the light cone of the first event (or that the first event is in the lower half of the light cone of the second event). If two events are outside of each other’s light cones there is no causal connection between them. This means that which one occurred first depends on what frame of references you’re in.

If there is a causal connection between two events they are “time-like” because one will occur definitively before the other. If there is no causal connection they are “space-like” because they definitively take place in different locations. In real life we do learn things from things that are causally connected to us because things that definitively happened in the past (are in the lower half of our light cone) are the only things that can effect us. This knowledge is called “memory”, something which Fraa Orolo likes to insult in Fraa Orolo’s Theory of Consciousness™ (see next section).

But talking about a strict causal connection from one universe to another is a bit silly. If there are parallel universes and say, there is a wormhole going from one universe to another, that’s a two-way connection. So the “acyclic” part doesn’t apply.

If an event occurs in the lower half of the light cone of the wormhole in Universe A and another occurs in the upper half of the light cone of the wormhole in Universe B then there is a causal connection from the first event to the second, but you could get the same sort of causal connection from an event in Universe B to an event in Universe A.

Think about this with planets. We’ve sent objects to Mars and retrieved objects from Mars. So there is a two-way causal connection from Earth to Mars. That’s because planets aren’t single points in space times, but entire world-tracks. The only way there can be a one-way causal connection from one planet to another is if the first planet was destroyed definitively before the other planet formed.

A model like this, would present a materialist view of a multiverse with no Hylæan Theoric World, be it on its own or in our head. It would also model the way we actually obtain information. But you can’t get that in Complex Protism.

It should also be noted, that during the course of this calca, they never mention the fact that causality is about time. This doesn’t come until the second time the mention Complex Protism several hundred pages later. This is also the only mention of isomorphism. One mathematician, Suur Moyra, is talking to another, Fraa Zh’avern, and says

“Uthentine and Erasmas were Thousanders, so their treatise was not copied out into the mathic world until the Second Millennial Convox.” She was speaking of the two avout who had developed the notion of Complex Protism. “Even then, it received scant notice until the Twenty-seventh Century, when Fraa Clathrand, a Centenarian—later in his life, a Millenarian—at Saunt Edhar, casting an eye over these diagrams, remarked on the isomorphism between the causality-arrows in these networks, and the flow of time.”

Erasmas the Elder and Uthentine were supposed to be from the Fourteenth Century, so it took 1300 years before anybody realized that being time-like had something to do with time. For a layman to not know that causality is all about time is understandable ignorance. For somebody who studies causality to not know that is stupidity. For the entire community of avout to take 1300 years to figure out something that should have taken a fraction of a second is stupidity beyond belief.

Also, stupidity beyond belief is Zh’avern’s reply:

“Isomorphism meaning—?” asked Zh’vaern.

Come on! All mathematicians know what isomorphism means! Even the ones who hate Category Theory know what isomorphism means! It’s one of the most basic concepts in mathematics! (Note: there is a spoiler about Zh’avern I won’t mention which some might consider to be an explanation for him not knowing that, but there is another spoiler I won’t mention which nullifies that).

Maybe if the people who wrote The Dictionary had actually taken the time to give a rigorous definition of causal domain it wouldn’t have taken them 1300 years to figure out the obvious.

On top of that, rather than viewing the connection between causality and time as being of utmost importance, its only used as part of a “How dare you accuse Complex Protism of being mystical mumbo-jumbo?!” attack on the Procians, and, as such, it’s completely ignored once everybody finishes their lunch.

And then we get to the issues of what was going through people’s head’s when they came up with, and elaborated on, Complex Protism. The developers of Complex Protism were Erasmas the Elder (not the narrator) and Uthentine, who looked at the picture of two boxes connected by an arrow and instinctively thought “that looks like a very primitive DAG” why not extend what ever it’s depicting to all DAGs.

But the two boxes with an arrow are also an example of a general directed graph, as well as a Category (boxes being the objects, and arrows being the morphisms). Why not extend it to those instead? Because their not acyclic and that would violate causality? But they’re universes, not events (see above).

If the reason you insist on the graphs being acyclic is because of causality, then why not look at posets instead? “Poset” is short for “partially ordered set”. A partial ordering is a relation that is reflecive (a ≤ a), anti-symmetric (if a ≤ b and b ≤ a then a = b) and transitive (if a ≤ b and b ≤ c then a ≤ c). The reason why it’s partial is because there could conceivably be two elements a and b such that neither a ≤ b nor b ≤ a.

For example, the positive integers have a partial ordering “|” where a | b if a is a factor of b. To see that this is partial note that neither 3 nor 5 are factors of the other.

Causality in both the Galilean sense and the Einsteinean sense is a partial ordering relation on spacetime. In the Einsteinean model there are more non-comparable elements than in the Galilean model. If you’re familiar with causality (like if you’re an avout who has devoted yourself to the study of causal relations between universes) this is pretty basic knowledge. So Erasmas the Elder and Uthentine should have at least thought of using posets to model the various Hylæan Theoric Worlds, like all other models of causality ever.

They could even spell it “pœsett”, thereby giving a little class to one of the most-mocked words in all of mathematics.

Also, DAGs are naturally related to posets. Every DAG has a poset structure on its vertices, where the partial ordering says a ≤ b if there is a path from a to b (you know like a causal relation). And every poset can be given the structure of a (possibly infinite) DAG where you put an arrow from a to b whenever a ≤ b.

But there are multiple DAGs that can give the same poset. Since the poset structure is what really matters with causality, you’d think that’s what Erasmas the Elder and Uthentine would be thinking about.

There are lots of other things that nobody seemed to consider? Is the model a lattice, meaning every pair of worlds has a least upper bound and a greatest lower bound? This would mean that in any two universes there is always some other universe that’s giving them both information. In some other model (like the “Firing Squad”) that won’t necessarily happen.

Is the poset/DAG finite or infinite? If it’s infinite, is there a “most Hylæan world” that has no Hylæan worlds of its own (if it’s finite there definitely is)? Also, if it’s infinite, is it continuous (like time), or discrete?

If it’s continuous the that should create a model of the multiverse that has extra dimensions representing the “Hylæan flow”. In this case that would create a new configuration space accounting for those other dimensions that one could do calculus on to quantify the rate at which information flows through the multiverse. If it’s discrete then information flows from one universe to the next via a quantum leap.

All of these different considerations would have a drastic effect on what the universes would be like, but nobody ever seems to think about that.

So, before moving on, let’s at least take care of one consequence of Complex Protism that is completely undisputable. If somebody can go from Universe A to Universe B, then there is a path in the DAG from Universe A to Universe B so Universe A is definitively “more Hylæan” than Universe B. Under no circumstances whatsoever under any model of Complex Protism could somebody move from a “less Hylæan” universe to a “more Hylæan” universe, because that would violate causality. If, hypothetically, someone was to go from a “less Hylæan” universe to a “more Hylæan” universe then that would mean Complex (and Simple) Protism are both utterly, flamingly wrong. So if, hypothetically, somebody claims to believe in Complex Protism and claims that somebody went from a “less Hylæan” universe to a “more Hylæan” universe, that that person would be completely and utterly full of shit in every way shape and form. And if, hypothetically, that full-of-shit person was the narrator and author surrogate of a 900+ page book who the reader was actually expected to like, then I’ve just wasted a good chunk of my life. But that could never happen.

Fraa Orolo’s Theory of Consciousness™

On a related note, in Anathem one of the defining differences between Halikaarnianism and Procisnism is that the Procians believe that the human mind is “just” a machine, and hence Artificial Intelligence is possible. Halikaarnianism, on the other hand, believes that consciousness is dependent on getting info from the Hylæan Theoric World, which is supposedly something that no machine can do. Therefore artificial intelligence is impossible.

So, back on the theme of finding consequences of theories that the characters themselves never considered. If consciousness is dependent on getting information from the Hylæan Theoric World, then in the Hylæan Theoric World you can’t get any new information, so there should be nothing to keep people from being “just” a machine.

In the Complex Protism view, people in the “more Hylæan” worlds get less information you can pick up, so the closer the people are to being “just” a machine. So even if somebody could go from the “less Hylæan” worlds to the “more Hylæan” worlds (which obviously can’t happen), they wouldn’t want to go to a world that would bring them one step closer to turning them into mindless automatons.

That is, if the Halikaarnian view of consciousness is correct, which it isn’t.

It should also be mentioned that, in real life, the Platonist Alan Turing believed in the possibility of artificial intelligence, because humans were machines anyway. Unlike Stephenson, Turing didn’t consider being called a machine to be a horrible insult.

Anyway, the Dumbledore-esque character Fraa Orolo develops the Halikaarnian view of conscious into a more detailed grand theory of consciousness that makes even less sense than the original version, one that combines the craziest most mystical aspects of Platonism and (oddly enough) formalism.

First of all, to elaborate on the original Halikaarnian view of consciousness. This is based on the idea that consciousness comes from neurons communicating to each other through millions of years of evolution. So far, that’s exactly the same as the Procian view, and the correct view. But the Halikaarnians’s add two differences, (1) the neurons communicate to each other through knowledge obtained from the Hylæan Theoric World and (2) rather than explaining rigorously how the human brain works they do it through a fable about a fly, a bat and a worm.

The fly-bat-worm fable itself isn’t too much of an issue, just a way of explaining it to laymen, but the Procian characters harp on that issue a lot more than the part that they would actually be expected to take issue with, namely the part about getting information from the Hylæan Theoric World.

When Fraa Orolo discusses it, he gives the example that looking at the front of a clothed person you can tell that they are also clothed on the back by the way their clothes hang. The explanation he gives is that humans have an innate sense of how clothes hang that they got from the Hylæan Theoric World.

The perfectly reasonable objection is made that seeing enough clothed people from different angles and other pieces of cloth hang gives us memories that we can use to figure this out. Orolo, through a bit of Hypotrochian Transquaestiation says “Thus far we have spoken only of the present. We’ve talked only of space—not of time. Now you would like to bring memories into the discussion.”

So instead of appealing to the past for information for his theory he proposes a model that creates a whole bunch of alternate universes that behave like the past but for whose existence there is no physical evidence. Therefore, the Hylæan Theoric World means that all people have an innate understanding of how the Sæcular World works (except for that sline who thought the sun went around Arbre, sor some reason).

Fraa Orolo uses this silliness as the basis for Fraa Orolo’s Theory of Consciousness™, by trying to figure out how the brain is able to get information from the Hylæan Theoric World. His answer: saying the word “quantum” over and over again.

You see quantum shit involves particles being in multiple states at the same time. One (perfectly legitimate but not universally agreed upon) theory is that those different states represent different universes. So the human mind, when thinking about a problem, creates a whole bunch of different universes through quantum shit and then settles on the one with the correct solution.

This is similar to quantum computing, and Orolo refers to a specific type of quantum computer that existed in the Anathem universe (called “Saunt Grod’s machine”) that had an unusually specific task. But this doesn’t really fit Orolo’s view.

First of all, the human mind doesn’t do a whole lot of quantum computation. The Heisenberg Uncertain Principle results in an issue called “quantum decoherence” whereby a quantum computer that has too much interaction with the rest of the world will collapse, so you need to keep a quantum computer isolated (physically isolated, not culturally isolated like the concents).

Second of all, quantum computing can’t accomplish anything that regular computing can’t accomplish. It can do certain things faster, like integer factorization (which means that quantum computers could break public-key cryptography if we can deal with quantum decoherence).

Third of all, quantum computing is every bit as formal and machine-like a process as regular computing. It was formalized by Paul Dirac through the well-defined notion of qubits as opposed to bits as the means of memory storage. There’s nothing there that says that bits of memory have to come from the past but qubits of memory have to come from a made-up universe that Plato pulled out of his ass.

Fourth of all, if you accept the “many worlds” interpretation of quantum mechanics, these many worlds don’t adhere to the “many worlds” interpretation of Complex Protism. There is not one quantum state that is “more Hylæan” or “less Hylæan” than another quantum state.

Fifth of all, a lot of Orolo’s theory is based on a misinterpretation of quantum physics that interprets “observation” as meaning “observation by a conscious being”. Orolo’s theory is really part of the “What the #$*! Do We Know?” school of quantum shit where you pretend that we can choose what quantum state you want to be in. It is also part of that school in that it involves regularly saying the word “quantum” over and over again without explaining how the quantum shit is supposed to work.

But this “What the #$*! Do We Know?” attitude is based on an extreme version of formalism where “there is no universal truth” is stretched to “there is no truth” to “we make our own truth?”

This makes the arguments with the Procians about consciousness especially weird. Since the Halikaarnian characters are adopting the worst aspects of both Platonism and formalism the Procian characters argue a more reasonable materialist view of consciousness.

In doing so the Halikaarnian arguments are based on the arguments that “intelligent design” advocates make in real life. Just like ID advocates claim that scientists current inability to create life means that life must have come from some divine source, the Halikaarnians argue that scientists current inability to create AI means that consciousness must have come from some divine source.

ID advocates accuse Darwinists of pulling a theory out of their ass and bending the evidence to fit that theory while being guilty of the exact same thing. The Halikaarnians do the same to the Procians.

Darwinism is backed by mounds upon mounds of evidence going back centuries to the point where a single bit of counter-evidence like “Oh yeah? Well how do you explain the eye?” isn’t sufficient to tear it down. Similarly, the “human mind as a machine” view of consciousness is backed by mounds upon mounds of evidence including the biology of the nervous system and various psychological tests, so a single piece of counter evidence like “How come we haven’t yet made a computer that can beat any human at Go?” is insufficient to tear it down. But the Halikaarnians, like ID advocates, pretend such evidence is sufficient.

And just as ID advocates like making populist-sounding declarations that present the Darwinists of being snotty elitists, the Halikaarnians make the same declarations of the Procians. The difference is that the Halikaarnians have complete elitist contempt for Sæcular world which makes their populist-sounding declarations ring even more hollow.

Real Life Consciousness

In real life scientists have actually studied how the brain works, and done psychological tests, and these would seem to confirm the view of consciousness promoted by the Procians (which is actually a materialist view, not a formalist view).

The brain and the nervous system are made up of specialized cells called “neurons”. This was discovered by dissecting humans and animals, not by staring at copper pots. Neurons have two ends, an “axon” and a “dendrite”. The axon of one cell is connected to the dendrite of the next. A neuron will receive a signal from some source through the axon and send the signal to the dendrite of the next. These signals are not vague, new age signals, but electrical and chemical synapses.

At one end you have sensory nerves that sense things, not from other neurons but from the outside world. These detect chemicals, temperature changes, light and other things and send initial signals. At the other end, you have motor nerves that are connected to muscles and create chemical reactions that cause the muscles to expand and contract.

These are Boolean operations, like in a computer. We have a pretty good understanding of how neurons work and we know that the brain is made of neurons linked together. There is no evidence of quantum computing and all the sensory nerves detect things in this world.

What we don’t know about the brain is precisely how the neurons are linked together. This means we know the brain is a machine, but we don’t know how the machine works. Even then, we do have some idea of how it works.

When talking about the Halikaarnian view of consciousness and Simple and Complex Protism, it is mentioned that Halikaarn claimed there was an “organ” in the brain that sensed particles truth particles, called “cnoöns” from the Hylæan Theoric World (in Complex Protism this would be the “more Hylæan” worlds).

Of course, there is no evidence that any such organ exists, nor any such particles. This is explained away by saying:

“But this was probably a translation error. Halikaarn was pre-Reconstitution, of course, so he was not writing in Orth but in one of the minor languages of his day. The person who translated his works into Fluccish did him a disservice by choosing the wrong word. Halikaarn wasn’t thinking of something like a gland. He was thinking of a faculty, an inherent ability of the brain, not localized in any one specific lump of tissue.”

So instead of being an actual organ, a legitimate scientific theory that is not true, we get a vague, ill-defined “faculty of the brain” that has no actual scientific meaning. Similarly cnoöns aren’t actual physical particles like protons, neutrons and electrons, but vague, ill-defined “universal truths” that never actually reach our universe but are just “perceived” by us in some vague, ill-defined way.

This is similar to the new age notion of “chi”. “Chi” is supposed to be a mystical energy. Supposedly chi can’t be measured because it’s energy. But other forms of energy, kinetic energy, nuclear energy, radiant energy, etc., can be measured, in units like “joules” and “calories”. People often count calories in food to make sure they’re getting the right amount of energy in their diet. But chi can’t be measured. This isn’t because it’s energy, but because it’s hogwash.

So while the Halikaarnians propose a model where a vague, ill-defined “organ” senses vague, ill-defined “particles” the real life-brain has real-life organs, actual lumps of tissue, called sensory receptors that sense actual particles, like photons and molecules.

We also have the time issue. Orolo dismissed the use of memory in his Theory of Consciousness™ because it was in the past and according to some sconic mumbo jumbo we do not perceive the past directly which for some reason he interprets to mean we do not perceive the past at all, so it can’t count as a given. What can count as a given and a legitimate source of information for the brain? The Hylæan Theoric World or, under complex Protism, the “more Hylæan” worlds.

But the perception issues with the past are also problems with the Hylæan Theoric World and “more Hylæan” worlds under those models. Moreover, they are both based on the same causal principles and are “isomorphic” as theories (not quite an isomorphism because two DAGs can have the same poset and not every DAG corresponds to a model of Einsteinean space-time, but they are “morphic”).

The difference between the “we get our information from the past” interpretation and the “we get our information from the Hylæan Theoric World” theories is that the former is backed by scientific evidence, while the latter is not. The Galilean and Einsteinean models of space-time were based on scientific evidence and have modern-day applications. The Galilean model isn’t strictly true because the speed of light is constant, but it’s applicable in the same sense as Saunt Hemn’s coordinates.

Furthermore the actual particles that our bodies sense through their actual organ, the sensory preceptors actually do come from the past. Furthermore, there is an actual lump of tissue in the brain, called the hippocampus that stores our memories.

By Occam’s Razor (Gardan’s Steelyard in the book), one would think that the theory of consciousness based on an actual organ that stores information from other actual organs that picks up actual particles that came from an actual place causally connected to the present would hold more water than one that’s based on a vague, ill-defined organ that picks up vague, ill-defined particles from a place causally connected to the present for whose existence there is no evidence.

Nonetheless, the Halikaarnians are willing to say the following of the Procians:

“But I would say that you have it backwards. You Procians have a theory—a model—of what consciousness is, and you make all else subordinate to it. Your theory becomes the ground of all possible assertions, and the processes of consciousness are seen as mere phenomena to be explained in the terms of that theory.”

Sorry, Halikaarnians, that’s what you’re doing.

One more thing about consciousness before getting to the plot. The Halikaarnians supposedly have a “smoking gun” in favor of their theory that AI is impossible (which is not actually a Platonist assertion). This “smoking gun” is that, since computers don’t know what their data represents it is possible to trick them, but that’s not the case with people.

Fun Fact: You actually can trick people.

SPOILER ALERT!!! THE REST OF THE REVIEW FOCUSES ON THE PLOT AND CONTAINS SPOILERS!!! IF THE REVIEW THUS FAR HAS CONVINCED YOU THAT ANATHEM IS A GREAT BOOK THAT YOU DON’T WANT SPOILED THEN READ NO FURTHER!!! BY WHICH I MEAN FEEL FREE TO KEEP ON READING!!!

Contact

The fact that Stephenson spends so much time on his mystical nonsense and awful Dialog is particularly disappointing because, when he focuses on the nuts and bolts of the plot Anathem is actually pretty good. At least it is until Stephenson’s Platonism and elitism gets the better of it.

The main plot is a first contact story similar to Carl Sagan’s Contact or Isaac Asimov’s The God’s Themselves. Like those two books, Anathem spends a great deal of time with characters working through data and discovering the existence of the aliens. In Anathem, we never actually see the aliens until two thirds of the way through the book.

Some of the best early scenes are when the characters realize that there is something strange in orbit around the planet. Specifically, something in a polar orbit, which doesn’t tend to occur naturally, so it’s thought to be some sort of a surveillance thing, with the perfectly accurate justification that a polar orbit gives, at some point in time, a view of every part of the planet.

They go through an arduous process to try and figure out what’s going on, with some false starts. When they finally find the satellite in question they make a camera obscura to see what happens when the satellite passes the sun. It then turns out that the satellite was making course adjustments when they were between the sun and the planet because they wouldn’t be caught on a telescope.

This part of the book was genuinely good and a welcome respite from all of the Hylæan Theoric World nonsense. It was one of the few instances when the characters displayed actual mathematical knowledge. Most importantly, the actions the characters took in these scenes followed naturally from the previous actions. Nobody behaved irrationally because the author was trying to make a point. It was like the early stages of an actual contact with aliens.

Later on, we get a scene very similar to one in Contact. This is after the aliens have made their ship visible. The characters can now view the ship with a telescope and see what it looks like, and they find a proof of the Pythagorean Theorem printed on the ship, which makes it clear that it’s piloted by intelligent life.

In Contact the aliens send a signal of beeps in prime numbers, which the humans use to determine that the source of the signal is intelligent. In both cases, the idea is that the math involved is too elaborate to occur naturally, but simple enough that it can be recognized.

Unfortunately, this is the first step in the deterioration of Anathem’s plot, as the characters interpret it as proof of the existence of the Hylæan Theoric World (and the Procians, as straw villains claim that it was faked rather than providing a legitimate counter-argument).

But if we go back to the legend of Cnoüs and his two daughters, we find that the lessons of the Pythagorean Theorem proof on the alien ship are the opposite of what Hylæa was arguing.

The supposed lesson of Cnoüs’s vision, according to Hylæa was that we shouldn’t confuse the symbol with the thing being symbolized. According to Hylæa, a picture of a triangle was a symbol and an abstract triangle in the Hylæan Theoric World was the “real” triangle. Just as the squares of cake or land are not really squares but just symbols of the “real” squares in the Hylæan Theoric world.

Also, just as lemons and bananas aren’t really yellow but just symbols of the “real” yellowness in the Yællœw Theoric World.

And yet it was not abstract “real” triangles on the alien ship, but pictures of triangles. What was on the alien ship was, according to the Hylæan interpretation, just “symbols” that represented the “real” triangles of the Hylæan Theoric World. And yet those “symbols” looked exactly like the “symbols” for triangles in the Anathem world and in our world.

How come all of these universes have the exact same “symbols” for triangle? Even on Earth, different parts of the planet have different symbols and different words and it’s only through communication that we come up with agreed upon things.

Perhaps, a picture of a triangle drawn on a space ship is not a “symbol” for an abstract “real” triangle, but the abstraction “triangle” is a property that is satisfied by the real (without quotes) shape drawn on the ship. I don’t know about Neal Stephenson and Plato, but for me, I usually think of real things as being real and made-up things as not being real, not the other way around.

What the different universes have in common is that the physics of these universes allow a large number of objects that can be modeled by Euclidean geometry, where the Pythagorean Theorem is true. If a universe existed where space-time was distorted beyond recognition people in that universe would think of either elliptical or hyperbolic geometry as being natural and would not consider the Pythagorean Theorem to be a universal truth. If there was a universe where space-time was Euclidean, but objects could only turn in right angles, they would consider taxicab geometry to be the natural geometry, in which the sum of the two sides of a right triangle is equal to the hypotenuse.

It’s worth looking a bit closer at Contact. This book has a similar story to Anathem and a similar mystical bent. It’s revered by astronomers and despised by mathematicians. The main reason mathematicians hate it is because of this atrocious bit of nonsense about the digits of π which is wrong on multiple levels.

Also, in the prime numbers bit in Contact, Sagan considered 1 to be a prime number, unlike most mathematicians. But, unlike the π bit, the prime numbers bit isn’t necessarily wrong, because there is some arbitrariness in the definition of prime numbers. As it’s commonly stated “numbers whose only factors are 1 and themselves” that would include 1, but mathematicians specifically rule out 1 as a prime in order to get the uniqueness of prime factorization.

But you could just as easily include 1 in your definition of “prime” and reformulate the statement of the uniqueness of prime factorization to be “unique up to multiplication by powers of 1”. There’s no reason to assume that the aliens would pick one definition over the other, but there is a reason why nobody would consider 12 to be prime or 3 to be not prime.

This is, first of all, because most objects in the universe are modeled by the integers (and not Z[sqrt(-2)]). If you have three objects, you can’t split them up into groups of the same size unless it’s one group or a group of one. If you have twelve objects, you can split it into three groups of four. So, other than 1, there wouldn’t be any disagreement.

But if there was a universe where there was a distinct notion of having sqrt(-2) objects, then they wouldn’t consider three to be a prime.

In the universes that appear in Anathem there are a lot more similarities than just math. It’s mentioned that hydrogen atoms are the same in both universes and oxygen atoms are very close (but carbon atoms are different). So there were already enough similarities to find a common ground without even needing math.

But if you can ignore the ramblings about the Hylæan Theoric World, the early stages of the first contact are some of the best parts.

Bad Science

The later stages, on the other hand, are all kinds of terrible. As are the middle stages.

Along the way, we encounter Fraa Orolo in exile after being anathematized for his discovery of the aliens. Instead of alerting the Sæcular world to what was going on, he sits in a temple on a volcano, isolates himself from the rest of the world and develops Fraa Orolo’s Theory of Consciousness™.

He claims that this is the most important thing he can do, because he had to “work with the givens”. But while he was hiding out on his volcano, he completely missed out on the spaceship making itself known, thus providing a whole lot more givens. He also claims that this is the best way to communicate with the aliens. Ultimately he draws an “analemma” on the ground, a shape based on the motion of the sun, and for some reason the aliens decide to land on the volcano.

No reason is given as to why this works beyond the “Neal Stephenson wrote the book so he had to prove the author surrogate characters right”. But in a non-fictional universe, it’s unlikely Orolo’s plan (or lack thereof) would have accomplished anything.

Once we actually meet the aliens we are treated to increasingly dubious science. It’s mentioned that the Arbre people can smell things on the alien ship, despite the fact that molecules are different in the other universes. This is explained by claiming that smell is caused by quantum quantum quantum quantum quantum. Oh yeah, did I mention quantum?

Actually smell is caused by a chemical reaction. The chemical reaction is determined by the shape of the molecules, which is determined by the structure of the atoms which involves some random quantum shit. But this is also true of digestion, and the Arbre people can’t digest the food of the other universes.

The “quantum scent” stuff would be forgivable if they didn’t use it as “proof” of Fraa Orolo’s Theory of Consciousness™. You see, because consciousness is based on quantum quantum quantum quantum quantum, and quantum quantum quantum quantum quantum involves parallel universes and the Arbre universe and the Hylæan Theoric World are parallel universes, it means that all conscious beings in different universes are bound together by quantum quantum quantum quantum quantum. This means that even in alternate universes, things still smell. And the fact that things still smell means that the Hylæan Theoric World must exist.

We get a similar explanation for why the aliens look like humans. The quantum quantum quantum quantum quantum that connects the universes meant there was a parallel evolution of conscious beings. One of the universes by the way, turns out to be our universe in the future, which includes plenty of other conscious animals that don’t look like humans (like dolphins). No explanation is given for why none of the aliens look like dolphins or why dolphins don’t look like humans.

It seems like things might get a little bit better. There is still the question of how these universes fit on the Complex Protism poset/DAG. It’s posed that they were once the same universe and then, in accordance with Fraa Orolo’s Theory of Consciousness™ some decision split the universes due to quantum quantum quantum quantum quantum.

The idea of a universe splitting into parallel universes due to quantum quantum quantum quantum quantum isn’t entirely beyond the realm of science. The idea that consciousness would be necessary for this is (and this idea is more in line with formalism than Platonism, despite being argued by the Halikaarnians). But ignoring the consciousness, it would give a good model for how they relate.

There would be a universe that is causally connected to all the parallel universes, or part of a universe. This would be the common past that the universes shared. This explanation (aside from the consciousness shit) makes sense. It has nothing to do with complex protism, since it’s the actual past and not a parallel universe that behaves like the past.

And this could even be used to explain some of the similarities between the universes. People look the same, not because of parallel evolution, but because much of the evolution occurred when the universes were the same.

Alas, it is not to be. They stick with the parallel evolution shit, but decide, arbitrarily that Complex Protism is right and they decide that the aliens are from the “Less Hylæan” universes and the Arbre people are in the “More Hylæan” universe!!!!!

So yeah, I wasted 900+ pages of my life reading this thing.

It gets worse. There are four universes worth of aliens who arrive, each going to the next. The Arbre people listen to their story and notice that the time in which they moved from one universe to the next corresponded to the “terrible events” and the three Sacks.

It’s also revealed that they moved from one universe to the next by traveling at great speed into a black hole, bending the light cones so much that they should have gone back in time but, (due to causality), it sent them to an alternate universe instead.

Problem. If the whole reason they ended up in an alternate universe was to preserve the causal relations of time, why did they flamingly violate the causal laws of Complex Protism (unless Complex Protism is nonsense)?

Problem. The Arbre people calculated the time between universe jumps by listening to the number of years the aliens gave and converting it to Arbre years. But the universe jumps involve heading at great speed into a black hole, based on the principles of Einsteinean relativity. But under Einsteinean relativity, time is relative, and the issues of time change drastically when one accelerates a lot. This is exactly the sort of situation where the twins paradox would figure prominently, yet they ignore it.

Problem. The human characters sing the praises of “Saunt” Gödel for noticing that if the universe is rotating then heading really fast into a black hole could let one travel back in time. But they never recognize “Saunt” Einstein for discovering the whole relativity thing in the first place. Maybe that’s because he’s an eeevil Procian. But that doesn’t explain why they don’t bother mentioning Gödel’s most important work, the Incompleteness Theorem.

It also turns out that Zh’avern isn’t really an avout from Arbre but a human agent named Jules Verne Durand. This could explain why he doesn’t know what “isomorphism” because he speaks a different language. But he’s also a linguist. Also nobody seems tipped off by the fact that an alleged mathematician doesn’t know what “isomorphism” means.

After another slight improvement with the action scene in orbit, the characters storm the spaceship in a scene that’s filled with improbable events explained by characters willing themselves by quantum quantum quantum quantum quantum to live in the universe where things worked out the way they wanted.

Bad science doesn’t necessarily make for bad science fiction. But in Anathem the bad science seriously hurts the book.

First of all, up until the aliens show up, the science is pretty hard. But once the aliens show up it rapidly breaks down. What started out as a Kim Stanley Robinson-level ultra-hard science fiction novel suddenly devolves to the level or realism of a Christian horror movie.

The other big problem, is that all the bad science is closely tied to the point the author is trying to make. You have a similar thing when comparing the Harry Potter books to the Narnia books. In the Harry Potter books the characters in the magic world all admit that magic exists, and the muggles don’t believe in magic until they witness it, and quickly become convinced whenever they do. In The Lion, the Witch and the Wardrobe the Professor chides Susan for not believing in Narnia on faith alone.

Anathem takes the Narnia approach. The bad science isn’t just a means to make the story-telling better (in fact, it makes the story-telling significantly worse), it’s actually trying to argue that the bad science is good science and that people who don’t believe in the bad science are eeevil Procians.

The Fulcrum and the Pedestal

Near the end of the book we learn more of the aliens’ culture, and boy does it suck. They are divided into two camps, called the “Fulcrum” and the “Pedestal”, which parallel the divides between the Procians and Halikaarnians and between the Mathic and Sæcular Worlds. Meaning the Pedestal are “good” and the Fulcrum are eeevil and stuuupid.

We meet Gan Odru, an alien general who launched a military coup in the space ship because he thought the lower ranking officers were too stupid, an act which started a civil war within the space ship. He’s the “good” guy.

The eeevil and stuuupid Fulcrum, you see, weren’t interested in the quest for the Hylæan Theoric World and (gasp!) wanted to focus on the well-being of the people in the ship.

But, just like the Procians, the Fulcrum are also eeevil, so they wanted to plunder Arbre for its resources. You see, because there exist fictional characters who want to help the well-being of their people and who are also eeevil means that anybody who wants to help the well-being of their people must be eeevil. That’s how you know that the Pedestal are right.

In this section, we not only see the cementing of Stephenson’s elitism, but also his association of math with religion. The Pedestal got their vision of the Arbre universe, similar to Cnoüs’s vision, but they interpreted it as a religious thing, like Deät, instead of Hylæa.

The eeevil Fulcrum thinks it was a hallucination and they are proven wrong. Not because there’s any precedent in real life for “visions” being anything other than hallucinations, but because Neal Stephenson said so. The moral of the story, trusting random visions is the exact same thing as engaging in rigorous reasoning through intense calculation. Provided that you’re willing to lead a military coup in defense of your beliefs.

Die Meistersinger von Saunt Edhar

The final chapter “Reconstitution”, is a rotten epilogue showing how everybody reacts to the events that transpired, and it bears an eerie resemblance to the end of Wagner’s opera Die Meistersinger von Nurnberg.

In Die Meistersinger von Nurnberg the Wagner surrogate character Walther von Stolzing, is trying to become a Meistersinger (master singer), but the eeevil bureaucrat of the Meistersingers, Sixtus Beckmesser, tries to keep him down by pointing out all of the musical errors he’s made. Meanwhile good Meistersinger Hans Sachs helps Stolzing out and instructs him to be himself.

This leads to a showdown singing contest between Stolzing and Beckmesser, where Beckmesser presents a technically correct but emotionless performance, and Stolzing sings from the heart winning everybody over despite not adhering to the strict rules.

At the end of the opera Sachs congratulates Stolzing but warns him that it’s important to recognize the musical tradition because it’s in danger of being destroyed by foreign influence so you have to honor your German Masters.

The same sort of thing is going on here. Throughout the book the Halikaarnians are complaining about the “discipline” in the concent that is imposed by the Procians through their trickery and the Sæcular World through the Sacks. In particular, they complain a lot about the fact that they’re not allowed to mingle with the Sæcular World.

So they get to the reconstitution, and things change. The concents are allowed to use all the banned technology, they aren’t even going to call them concents anymore. And they can mingle with the Sæcular World whenever they want to (which primarily means selling them souvenirs). But there’s still a wall around the Mathic world.

When a Sæcular individual, Artisan Quin, asks Erasmas why they’re keeping the wall, he tells them it’s of symbolic value. But as the narrator, he tells the reader:

“But I knew I was not being altogether forthright. Half a mile away I could make out half a dozen people in bolts, peering through instruments and pounding in stakes: Lio and the crowd of ex-Ringing Vale avout he ran with now. I knew exactly what they were discussing: when war breaks out between the Magisteria and we plug the holes in the wall with gates, we’ll want interlocking fields of fire between this bastion and the next to repel any assault on the intervening stretch of wall…”

So, it’s okay to sell souvenirs to the Sæcular World, but you’re still going to kill them when the time comes.

It’s even nastier when you look at the other aspects of the “Reconstitution”. We see the nü-concents seizing land from the Sæcular World (which is “okay” because now the Mathic World has private property). There’s also a new four-war government in which the Sæcular World, Mathic World, Pedestal and Fulcrum each get equal representation. This despite the fact that the Fulcrum is bigger than the Pedestal and the Sæcular world is way bigger than the Mathic World. So much for democracy.

And all of this, for some reason, is considered good. And the ending is supposed to be “happy”. Sorry, I’m not buying it.

What Could Have Been

The real shame is, this could have been a good book. The early plot was great, as were the descriptions of architecture and landscapes. And the arctic adventure was pretty good too, if you can get past the ninjas and the fact that the sled-guy was supposed to be a good guy.

If he had stuck in the Contact/The Gods Themselves mode and done a straight first contact story without the Hylæan Theoric World stuff, it could have been great. Keep most of the action scenes, but replace the bad science with good science, the horrible philosophizing with good philosophizing and the egregious politics with good politics. Just get rid of 700 pages and replace them with 700 different pages.

It's a world where a combination of Tea Party-"populism" and Randian elitism has forced a separation between the scientists and the rest of the world. It hasn't been going on for 3689 years, but a pretty long time nonetheless. The scientists don't dress like monks, but some of them have become so isolated from the real world that they turn to an elitist pseudo-scientific mysticism.

Then aliens come and a small group of scientists discover them (going through the exact same process as in the book), and realize this is going to be an Arbre-shattering event. They have no choice but to leave the Univærsitie and enter the Lib'rahl Ærts Majœr World, to alert them about the aliens. Along the way they have an arctic adventure and are shocked by how horrible the sled-guy is but feel relieved when they completely and utterly fail to encounter any ninjas.

They have to argue against both the Glenn Beckian anti-intellectualism of the Lib'rahl Ærts Majœr World and the elitist mysticism of the Univærsitie. Maybe they'll also encounter a Ron Paul figure who supports both anti-intellectualism and elitism and they all get very confused as to how anyone could take that guy seriously. In doing so they have some non-horrible philosophical discussions about math and go to meet the aliens via an action scene in orbit (exactly like the one in the book). And it ends without completely abolishing democracy! This thing practically writes itself!

Even if he really wanted all that Hylæan Theoric World nonsense he could have done that well. The fact that vampires don’t exist doesn’t prevent people from enjoying Buffy the Vampire Slayer. The fact that magic doesn’t exist doesn’t prevent people from enjoying the Harry Potter books. And the fact that God doesn’t exist doesn’t prevent people from enjoying the short-lived NBC series Kings.

That’s because those are all fantasy series. He could have kept the Platonism and the religious imagery if he had just made it into a fantasy novel. It could have been called Fraa Erasmas in Mathmagic Land.

And since he would no longer be bound by trying to give scientific credence to the Hylæan Theoric World, he could dump all the Complex Protism and the quantum quantum quantum quantum quantum. Erasmas could actually go to the Hylæan Theoric world and have conversations with “pure” triangles. It would be something like Edwin A. Abbott’s Flatland, but with monks.

It would have also allowed a lot more focus on actual math.

Or if he wanted to go for a fantasy based on formalist instead of Platonist mysticism he could have done something like Tarsem Sing's movie The Fall, but with Roy Walker reading Alexandria a math text book.

Even as science fiction, he could have gotten away with some of the bad science. There are plenty of science fiction stories that have faster-than-light travel, violations of causality and a tendency to say the word “quantum” over and over again in meaningless technobabble. But those stories (a) don’t have any pretense to being hard science fiction and (b) don’t take their bad science seriously.

Regardless of what he could have done, it would have to get rid of the universal straw men, constant Crichton-speak, contempt for non-scientists, contempt for science and love of military dictatorships.

But that’s what Neal Stephenson believes in, so for Anathem to be good, it would have pretty much had to have been written by someone else.

Otherwise we end up with the same problem: it’s a great novel of ideas where the only good bits are the action scenes. As talented as Neal Stephenson is as a writer, the rot is in the foundations.

12 comments:

"So if, hypothetically, somebody claims to believe in Complex Protism and claims that somebody went from a “less Hylæan” universe to a “more Hylæan” universe, that that person would be completely and utterly full of shit in every way shape and form. And if, hypothetically, that full-of-shit person was the narrator and author surrogate of a 900+ page book who the reader was actually expected to like, then I’ve just wasted a good chunk of my life. But that could never happen."

"a lot of Orolo’s theory is based on a misinterpretation of quantum physics that interprets “observation” as meaning “observation by a conscious being”. Orolo’s theory is really part of the “What the #$*! Do We Know?” school of quantum shit where you pretend that we can choose what quantum state you want to be in."

Man, I hated that movie. I hate even more that everyone seems to walk away from it thinking it's real science.

"This is similar to quantum computing, and Orolo refers to a specific type of quantum computer that existed in the Anathem universe (called “Saunt Grod’s machine”) that had an unusually specific task."

The task referred to is what we on Earth call the 'traveling salesman problem', which is NP-complete. If you could solve this seemingly very specific problem efficiently, you could use your algorithm to solve tons of other optimization problems efficiently.

Stephenson seems to be suggesting that quantum algorithms can solve NP-complete problems quickly. Here on Earth, it's very much an open question whether this is true (in complexity theory terms, whether BQP contains NP), just as we don't know whether ordinary computers can solve NP-complete problems quickly (this is the famous "P vs NP problem").

It's a common misconception that quantum algorithms can easily solve NP-complete problems. It's unclear from the text whether Stephenson suffers from this misconception. However, I am annoyed that he chose the traveling salesman problem, which on Earth is complexity theorists' favorite example to explain the P vs NP problem to laypeople. Wouldn't they have a different favorite example on Arbre?

Can't believe I just read most of that post. :) Probably silly to comment on it too, but oh well.

The way I interpreted the complex protism DAG was that the arrows represented a specific type of information flow, one which presumably causes circles and whole numbers to exist in our Universe and allows us to recognize them. There are apparently other connections which allow travel in potentially the opposite direction, but the DAG isn't meant to specify them.

Also, not everyone favors posets over dags when modeling causal relationships. Judea Pearl and other 'structural equation' type causal theorists see the dag as more fundamental. Even the older 'possible worlds' views define the idea of 'direct causes' first and then make it transitive.

Oh one more thing. Physicists have of course been putting larger and larger objects into quantum superposition. The Incanters in the book apparently have (subtle, implausible) methods of putting very large objects into superposition. I found it especially nice how the most nonsensical acts of superposition occurred in orbit, where the people weren't interacting with very much else - especially after destroying their radio so they wouldn't interact with the surface.

What Fraa Jad did seemed a lot like wishing a particular reality into existence, but the way I ended up understanding it was that he caused several possible realities to interfere with one another, which of course is what one wants a quantum computer to do. So I didn't find it completely impossible.

It seems easy to accommodate the Parallel Postulate in a platonic view of some sort: we know that the parallel postulate is true on some types of surfaces and not others, so the "true" axioms can describe surfaces and let the various geometries arise on separate objects. One could at least hope a similar situation holds for other independent axioms.

It seems plausible there could be universes where approximate instantiations of Euclidean geometry rarely emerge, but it is much harder to imagine a universe where there is a notion of having, as you say, sqrt(-2) of something. It seems at least slightly plausible that such a universe simply doesn't exist; that is, perhaps there is no (consistent) physics with both that property and the right sort of complexity to allow life to observe that property. So it is at least slightly plausible, that there may be certain mathematical objects instantiated in nearly any Universe with enough computational capability to observe its own properties in some manner.

Well, I found the book entertaining, so that's fine by me. I agree that libertarians don't appear to have thought things through and get all pissy when you point this out, but who cares? They're fun to argue with.

Small technical point: Smell may well be cause by vibration rather than shape. See Luca Turin's 'vibration theory of olfaction' esp. the section on borane.

The author's politics are a "strike against all of his books"? Are you kidding me? You can't separate the work from the author? Guess we all have to hate Bach since he was an alcoholic or Beethoven since he beat his kids.